Queuing Tutorial

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Queuing Rule of Thumb

After meeting most of my clients, I realized that to explain standard queuing theory to layman require considerable herculean efforts. I feel that my whole mathematic, statistics and operation research knowledge are useless when I can see their blank face saying can you explain that in easier manner, please? Definitely I need to explain my knowledge in much simpler language. Thus, I need simpler way to compute queuing.

Queuing theory has been developed since the beginning of last century (by A. K. Erlang in 1917). It is quite surprising that the theory did not become common public knowledge. Despite all the advancement of the queuing theory in almost a century, many people still have lack of understanding about queuing. In our daily life, we still face a lot of queuing from school, to bank and from restaurant to toilet. We still are being frustrated with our daily traffic and a long queue in registration and security check. In fact, the queue does not diminish by the advancement of our knowledge in queuing theory but it grows into more complex reasons. Public and business understanding of queuing is still very far from the science. For instance, instead of distributing the queue over time and space, many managers do the very common mistake by making policy to concentrate the demand over time and space. Instead of solving the queuing problem, they create more severe queues.

Ignorance of the management regarding this minimum number of servers lead to many trouble. I have experienced attending a conference in a luxury hotel where the number of table was inadequate and lead the participants to queue for more than 2 hours. I have observed many schools ignore the simple computation of number of servers during school registration. I have seen too many government offices around the world set up many serial servers instead of parallel servers. They don't realize that regardless how many servers do you have, if you set them in series, they are basically behave similar to only one server. There are so many unnecessary queues with so much frustration can be solved if they want to learn very simple formula of queuing rule of thumb.

Hall (1991) cited an argument of his previous paper that operation research profession could and should be more scientific and less mathematical. In fact, Hall also suggested another queuing rule of thumb that queues should be small if the number of server is

Hall's Queuing Rule of Thumb

s = number of parallel servers
Traffic Intensity = traffic intensity = ratio of mean arrival rate and mean service rate.

Here is my queuing rule of thumb (note: I have published my finding about queuing rule of thumb in scientific journal and conferences in 2012 ):

Teknomo's Queuing Rule of Thumb

s = the number of parallel servers (servers)
N = total number of customers (customers)
r = service time (seconds/customer/server)
T = max time to finish the queue (seconds)

If you use the equal sign instead of larger than or equal, the queuing rule of thumb that I proposed above will give the absolute minimum number of servers at the worst quality of service . One cannot go lower that that number of server without either reduce the demand, prolong the service or increase the service time.

You can use the queuing rule of thumb calculator below to determine the number of minimum number of servers, given the total number of customers, service time and the service duration.

customers seconds seconds


In a conference lunch usually people can take their own food. The parallel servers in this case are the side of the serving tables. If each conference participant need about 45 seconds to take his/her food and there are 1000 participants, how many tables should you provide such that the lunch break can be finished within 1 hour?

Solution :

We set r = 45 seconds and N = 1000 customers and set T = 3600 seconds. Using the queuing rule of thumb above, you see that at minimum, you need s > 1000 * 45 / 3600 = 12.5. However, if you can use two side of the table, in this case, you only need s > 12.5 / 2 = 6.25. We need to round up to a whole number since the number of server is discrete. Thus, 7 serving tables with two sides must be the minimum number of tables to serve the worst level of service.


A school of 10,000 students set certain days for registration and tuition fee payment (a serial servers to get registration form, check all delayed payments, pay at cashier and get certificate of payment). One working day is from 8:00-17:00 minus 1 hour lunch break. Each parent needs about 36 seconds to be served by all the serial servers. How many days is needed to register all students?

Solution :

Due to serial servers, the actual server is only s=1.

N = 10,000

t = 36 seconds

One day is 8 hours = 8 * 3600 = 28,800 second/day

Teknomo's Queuing Rule of Thumb Example seconds

Registration days needed = 360,000/28,800 = 12.5 days rounded up to 13 days to serve at the worst level of service.


During morning peak hour (6:00-7:00), approximately 4500 cars will enter a favorite school to drop their children to elementary school. Each drop point requires approximately 60 seconds (including maneuver time and bring down the heavy luggage). Each car requires about 6 meters to stop & maneuver. How long should the minimum dropping line?

Solution :

N = 4500 cars

T = 60 minutes

r = 1 minute

Server space = 6 meter.

Teknomo's Queuing Rule of Thumb example parallel dropping line

The meander for dropping line should be at least 75 * 6 meter = 450 meters to serve the worst level of service. If the dropping line is less than 450 meters or if the dropping line are not operated in parallel, the children will be late for school. (note: the term dropping point is incorrect because it makes the designer to think in term of a point rather than a line. It is interesting that the servers here the dropping line measure in meter rather than human server in traditional queue).

Compared to the standard queuing formulas, the queuing rule of thumb is imprecise and probably too simple. My goal in introducing this simple formula is to explain queuing to public in a very simple manner. Using what I called as queuing rule of thumb above, we should be able to explain to elementary school students about queuing theory. It is my dream that that one day, this type of simple queuing theory would be part of grade school or elementary school curriculum. The lack of understanding on how to solve queuing requires public education.

We should concern with how the system behave and less concern with abstract symbol manipulation. To understand queuing theory fully, one need to have background in statistics and differential equations and to be able to manipulate Markov chain. Looking at the recent development of vacation queuing theory and queuing network, the trend of queuing theory development is toward more precision with requires higher mathematical manipulation. While higher precision queuing theory computation is great for the body of knowledge, however this trend also produces wider gap between the theory and practice. We cannot educate queuing theory as public common knowledge if the required background knowledge of the mathematics would make even a post-doctoral fellow in mathematics several days to understand. We really need a much simpler version of the queuing theory that can be used to educate public about queuing. Hence, queuing rule of thumb formula is needed.

Compared to the actual queuing formula, the queuing rule of thumb only give you rough approximation. However, its simplicity is very handy to compute the need of the number of servers without the need to understand the concept of probability. The purpose of the rule of thumbs is not to gain precision or optimization. The aim is to gain understanding of the current queuing situation and to give the layman simple tools to solve queuing problem more creatively.

In what follows in this tutorial, you will be introduced with more technical details of the most common standard formulation of queuing theory.

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These tutorial is copyrighted .

Preferable reference for this tutorial is

Teknomo, Kardi. (2014) Queuing Theory Tutorial